cp's OEIS Frontend

A019269 Let Dedekind's psi(m) = product of (p+1)p^(e-1) for primes p, where p^e is a factor of m. Iterating psi(m) eventually results in a number of form 2^a*3^b. a(n) is the number of steps to reach such a number.

%I A019269 #25 Jan 18 2020 08:47:36
%S A019269 0,0,0,0,1,0,1,0,0,1,1,0,2,1,1,0,1,0,2,1,1,1,1,0,2,2,0,1,2,1,1,0,1,1,
%T A019269 1,0,3,2,2,1,2,1,2,1,1,1,1,0,2,2,1,2,1,0,1,1,2,2,2,1,2,1,1,0,2,1,2,1,
%U A019269 1,1,1,0,4,3,2,2,1,2,2,1,0,2,2,1,1,2,2,1,2,1,2,1,1,1,2,0,3,2,1
%N A019269 Let Dedekind's psi(m) = product of (p+1)p^(e-1) for primes p, where p^e is a factor of m. Iterating psi(m) eventually results in a number of form 2^a*3^b. a(n) is the number of steps to reach such a number.
%C A019269 a(A019268(n)) = n and a(m) <> n for m < A019268(n). [_Reinhard Zumkeller_, Apr 12 2012]
%D A019269 Peter Giblin, "Primes and Programming - an Introduction to Number Theory with Computation", page 118.
%D A019269 R. K. Guy, "Unsolved Problems in Number Theory", section B41.
%H A019269 Reinhard Zumkeller, <a href="/A019269/b019269.txt">Table of n, a(n) for n = 1..10000</a>
%t A019269 psi[n_] := Module[{pp, ee}, {pp, ee} = Transpose[FactorInteger[n]]; If[Max[pp] == 3, n, Times @@ (pp+1)*Times @@ (pp^(ee-1))]];
%t A019269 a[n_] := Length[NestWhileList[psi, n, FactorInteger[#][[-1, 1]] > 3&]] - 1;
%t A019269 a /@ Range[99] (* _Jean-François Alcover_, Jan 18 2020 *)
%o A019269 (Haskell)
%o A019269 a019269 n = snd $ until ((== 1) . a065333 . fst)
%o A019269                         (\(x, i) -> (a001615 x, i+1)) (n, 0)
%o A019269 -- _Reinhard Zumkeller_, Apr 12 2012
%Y A019269 Cf. A001615, A065333.
%K A019269 nonn
%O A019269 1,13
%A A019269 _Jud McCranie_